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Mathematically, '''continuum''' can refer to the [[Real_line|real line]], its [[Cardinality|cardinality]] <math>\mathfrak{c}</math>, or any [[Continuous|continuous]] [[Connected_(topology)|connected]] [[Dense_subset|dense]] [[Linear_order|linear order]]. Generally, when mathematicians say "the continuum", they are referring to one of the first two possibilities (context clarifies which one). The [[Continuum_hypothesis|Continuum Hypothesis]]  conjectures that there is no set of  cardinality bigger than that of the [[natural number]]s <math>\aleph_0</math>, but smaller than the cardinality of the set of the real numbers <math>\mathfrak{c}</math>.

Mathematically, '''continuum''' can refer to the [[Real_line|real line]], its [[Cardinality|cardinality]] <math>\mathfrak{c}</math>, or any [[Continuous|continuous]] [[Connected_(topology)|connected]] [[Dense_subset|dense]] [[Linear_order|linear order]]. Generally, when mathematicians say "the continuum", they are referring to one of the first two possibilities (context clarifies which one). The [[Continuum_hypothesis|Continuum Hypothesis]]  conjectures that there is no set of  cardinality bigger than that of the [[natural number]]s <math>\aleph_0</math>, but smaller than the cardinality of the set of the real numbers <math>\mathfrak{c}</math>.

The continuum is called so because it was the first (and most prominent) [[Continuous|continuous]] set studied by mathematicians. No additional numbers may be added to the continuum (real line) without losing its dense linear order without [[endpoint]]s. Since the [[Complex_number|complex numbers]] add the [[Imaginary_number|imaginary number]], i, to the real line, this is one reason they have no natural linear order.

The continuum is called so because it was the first (and most prominent) [[Continuous|continuous]] set studied by mathematicians. No additional numbers may be added to the continuum (real line) without losing its dense linear order without [[endpoint]]s. Since the [[Complex_number|complex numbers]] add the [[Imaginary_number|imaginary number]], i, to the real line, this is one reason they have no natural linear order. Any linear order of the complex numbers needs the [[Axiom of Choice]] to be constructed.

[[Category:Mathematics]]

[[Category:Mathematics]]